Remarks on Differentiable Actions of Non-Compact Semi-Simple Lie Groups on Euclidean Spaces

Abstract

Introduction. It is well known that differentiable actions of compact Lie groups are far more regular than those of the non-compact Lie groups and the behavior of compact Lie group actions are comparatively much more well understood than those of the non-compact Lie groups. Of course, one of the most basic and important cases of non-compact Lie group actions are those Rl-actions which are naturally associated with ordinary differential equations [6 ]. So far, much important research has been done in that direction. However, knowledge about R'-actions still seems rather inadequate. Since the group R1 is a simply connected one-dimensional Lie group and it contains no non-trivial compact subgroups, neither the theory of Lie groups nor knowledge about compact Lie group actions are of any help in the study of R'-actions. In this note, we try to look at the other extreme case of actions of large semi-simple non-compact Lie groups. The situation here is quite different from the study of RI-actions. Roughly speaking, the problem is more complicated on the one hand, and has much more built in structures on the other hand. For example, one naturally expects that the profound theory of semisimple Lie groups and the knowledge of compact Lie grfoup actions might be useful at least for some special but natural cases. Due to the fact that one knows so little about the behavior of differentiable actions of non-compact Lie groups in general, it seems to be worthwhile to investigate some special but natural cases such as semi-simple Lie group actions on euclidean spaces. Let G be a semi-simple non-compact Lie group. It is well known that there exists a maximal compact subgroup H which is unique up to conjugacy. For a given differentiable manifold M'4, let r(GMm]Ii4) and r (H, Mm) be the set of equivalence classes of differentiable actions of G and H on Ml', respectively. To restrict those G-actions on MIlm to H induces a natural luap i !: r (G, Mm) r (H, Mm) which is independent of the choice of the maximal compact subgroup H. A reasonable way to study G-actions on Mm is to investigate the image and the kernel of the above map. To be more precise,